MathsCBSE Class 12 · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2025-26: question 27

From the official CBSE sample question paper for 2025-26, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerCompetency-basedApplication of Derivatives

A spherical ball of ice melts in such a way that the rate at which its volume decreases at any instant is directly proportional to its surface area. Prove that the radius of the ice ball decreases at a constant rate.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

Open one hint at a time and try again before the next.

Hint 1

Translate 'rate of decrease of volume proportional to surface area' into an equation with a constant $k$ .

Hint 2

Differentiate $V = \frac{4}{3} π r^{3}$ w.r.t. $t$ and compare with $4 π r^{2}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let $r$ be the radius at time $t$ . Volume $V = \frac{4}{3} π r^{3}$ …(1), surface area $S = 4 π r^{2}$ …(2).
½ mark
Given the volume decreases at a rate proportional to $S$ : $\frac{d V}{d t} = - k S$ , $k > 0$ a constant …(3).
½ mark
Differentiating (1) w.r.t. $t$ : $\frac{d V}{d t} = \frac{4}{3} π \cdot 3 r^{2} \frac{d r}{d t} = 4 π r^{2} \frac{d r}{d t}$ …(4).
1 mark
From (3) and (4): $- k S = 4 π r^{2} \frac{d r}{d t} = S \frac{d r}{d t}$ [using (2)].
½ mark
$\Rightarrow \frac{d r}{d t} = - k$ , a constant. So the radius decreases at a constant rate. Hence proved.
½ mark

Answer: $\frac{d r}{d t} = - k$ (constant), so the radius decreases at a constant rate

Where the marks are

Writes $V$ and $S$ in terms of $r$ (½ mark)
Forms $\frac{d V}{d t} = - k S$ (½ mark)
Differentiates $V$ to get $4 π r^{2} \frac{d r}{d t}$ (1 mark)
Equates using $S = 4 π r^{2}$ (½ mark)
Concludes $\frac{d r}{d t} = - k$ (½ mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)
Topic: Rate of Change of Quantities

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 27

Hints

Step-by-step solution

Revise the chapter

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